Determine what Decartes Rule of Signs says about the number of positive and negative real roots for the polynomial function P(x)=8x^3 +2x^2-14x+5

Answer:

Step-by-step explanation:

the green one no

the red ones yes

The value of the equation is x = 4 and x = -5 ± √10i

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , the value of A is

Substituting the values in the equation , we get

( x - 4 ) / ( x + 5 ) + 9 / ( x² + 11x + 30 ) = 1/( x + 6 )

On simplifying the equation , we get

( x - 4 ) / ( x + 5 ) + 9 / ( x + 5 ) ( x + 6 ) = 1/( x + 6 )

Multiply the equation with the LCM , we get

( x - 4 ) ( x + 5 ) ( x + 6 ) + 9 ( x - 5 ) = ( x - 5 ) ( x + 5 )

x³ + 7x² - 14x - 120 + 9x - 45 = x² - 25

Subtracting x² on both sides of the equation , we get

x³+ 6x² - 5x - 165 = -25

Adding 25 on both sides of the equation , we get

x³+ 6x² - 5x - 140 = 0

Hence , the solution to the equation is x = 4 ,  -5 ± √10i

So , the larger solution is x = 4

And , the smaller solution is x = -5 ± √10i

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-2W + 6 is The opportunity cost of 1 coconut is ½ bundle of firewood.

Any mathematical statement that includes numbers, variables, and an arithmetic operation between them is known as an expression or algebraic expression. In the phrase 4m + 5, for instance, the terms 4m and 5 are separated from the variable m by the arithmetic sign +.

For Alice:

In the amount of time it takes to gather a bundle of firewood, Alice could have gathered (1/2) * 6 = 3 coconuts. Thus, the opportunity cost of one bundle of firewood is 3 coconuts.

The opportunity cost of 12 coconuts is 4 bundles of firewood.

Alice can gather 12 coconuts in 8 hours. If she were to gather firewood instead during these 8 hours then she would be able to gather 4 bundles of firewood

Therefore, when she chooses to gather 12 coconuts she is giving up 4 bundles of firewood.

Denoting coconuts through C and firewood through W, we are able to discover the equation for the PPF through graphing directly, or through noting that if all time is spent amassing wood, she will be able to accumulate four bundles of wood, so the W intercept is four.

Since the opportunity cost of a bundle of wood is 3 coconuts, the coefficient on C will be -1/3.

Therefore, we have W = (-1/3)C + 4, or equivalently C = -3W + 12.

(Alice’s PPF is plotted below in the solution to part (c).)

For Bob:

Since Each bundle of firewood takes as long as gathering 2 coconuts, the opportunity cost of one bundle of firewood is 2 coconuts.

The opportunity cost of 1 coconut is ½ bundle of firewood. Repeating the procedure from before, we find Bob’s PPF can be expressed by either W

= (-1/2)C + 3 or C = -2W + 6.

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Complete question:

Alice has found herself on a desert island and must gather supplies to survive. Alice has 8 hours of useful stamina that she can use towards gathering firewood or coconuts. She finds that she can gather 1 bundle of firewood every two hours, or 6 coconuts every 4 hours.

a. In terms of the number of coconuts, what is the opportunity cost of 1 bundle of wood? What is the opportunity cost of 12 coconuts in terms of bundles of wood?

b. Draw a plot of and give an algebraic expression for Alice’s production possibility frontier (PPF). Measure coconuts on the horizontal axis and bundles of wood on the vertical axis.

Bob just crash-landed on the island too. He doesn’t have quite the stamina that Alice does so he can only work 6 hours a day. He finds that he can gather a bundle of wood in 2 hours or 1 coconut per hour.

Let the width of the path be represented by x.

The total width of the pool and the path would then be 16 + 2x (for the two shorter sides) and 28 + 2x (for the two longer sides).

The area of the pool itself is 16 × 28 = 448 square meters.

The area of the combined pool and path is 864 square meters.

So the area of just the path is:

864 - 448 = 416 square meters

The area of the path can also be expressed as the area of the larger rectangle (16 + 2x) × (28 + 2x) minus the area of the pool (16 × 28).

So we can set up the equation:

(16 + 2x) × (28 + 2x) - 448 = 416

Expanding the left side and simplifying:

4x^2 + 88x - 480 = 0

Dividing by 4:

x^2 + 22x - 120 = 0

Using the quadratic formula:

x = (-22 ± sqrt(22^2 - 4(-120)))/2

x = (-22 ± 26)/2

x = 2 or x = -24

The width of the path cannot be negative, so the width of the path is 2 meters.

Answer:

164

Step-by-step explanation:

c = number of caps

t = number of t-shirts

c + t = 278

8c + 7t = 2110

   -7c - 7t = -1946

+   8c + 7t = 2110

-------------------------

      c         = 164

Answer: 164

The relationship between x and y is y = x + 2

From the question, we have the following parameters that can be used in our computation:

(1, 3), (2, 4), (3, 5) and (5, 7)

See attachment for the line

Using the above as a guide, we have the following:

The three other points are (6, 8), (7, 9) and (8, 10)

From the ordered pairs, we can see that

y is the sum of x and 2

This means that

y = x + 2

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Due to length restrictions, we kindly invite to see the explanation below to know the complete truth table of the composite proposition.

According to logics, propositions are constructions that contains a truth value and in this question we must complete a truth value of a composite proposition, that is, a combination of simpler proposition.

To complete a this table, we must begin with the most simple proposition and go through propositions with more complexity:

p     q      r       ¬ p      ¬ q      p → q      r ∧ ¬ p      r ∨ ¬ q

T     T      T         F         F         T             F               T

T     T      F         F         F         T             F               F

T     F      T         F         T         F             F               T

F     T      T         T         F         T             T               F

F     F      T         T         T         T             T               T

F     T      F         T         F         T             F               F

F     F      F         T         T         T             F               F

[(p → q) ∨ (r ∧ ¬ p)]       [(p → q) ∨ (r ∧ ¬ p)] → (r ∨ ¬ q)

           T                                          T

           T                                          F

           F                                          T

           T                                          F

           T                                          T

           T                                          F

           T                                          F

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(-2, 64), (-1, 49), (0, 36), (1, 25) and (2, 16) are the ordered pairs of given equation.

Two or more expressions with an Equal sign is called as Equation.

The given equation is y=x²-12x+36

y equal to x square minus twelve times of x plus thirty six.

Now let us find some values of x and y.

If x=-2

y=64

If x=-1

y =49

If x=0

y=36

If x=1

y=25

If x=2

y=16

Hence, (-2, 64), (-1, 49), (0, 36), (1, 25) and (2, 16) are the ordered pairs of given equation.

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Answer:

x - 6,5,7,9,4

y - 0,1,1,9,4

Step-by-step explanation:

The number of animals left in the wild after 4 years is approximately 306.

Compound interest is defined as the amount of interest which has been calculated on the principal amount as well as the amount accumulated over the previous period is also included.

We are using the concept of compound interest to solve the given problem.

The formula to find the final amount in compound interest is,

A = P (1 + r) ^t

Here P = 400

r = -6.5% = -0.065 (negative sign because the rate is decreasing)

t = 4 years

Substituting the values,

A =400 (1 - 0.065)⁴

  = 305.71 ≈ 306

Hence there will be 306 animals left in the wild after 4 years.

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The statement that is always correct regarding the multiplication of x and z is given as follows:

The multiplication of x and z is always negative.

The multiplication signal rules is defined by the terms two by two, and the rule is given as follows:

For this problem, we have that x and z have different signs, hence the product is going to be always negative.

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The distance between each pair of letters in the word is given by the equation A = 5 inches

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the distance between each pair of letters be A

Now , the equation will be

Let the total distance of the wide banner be = 61 inches

Now , the word is given by { E , A , G , L , E , S }

The number of letters in the word = 6 letters

The width of each letter = 6 inches

So , the total length of all the letters = 6 x 6 = 36 inches

The remaining distance in the wide banner = total distance of the wide banner - total length of all the letters

On simplifying the equation , we get

The remaining distance in the wide banner = 61 - 36

The remaining distance in the wide banner = 25 inches

Now ,

The distance between each pair of letters = remaining distance in the wide banner  / 5 spaces

On simplifying the equation , we get

The distance between each pair of letters A = 25 / 5

The distance between each pair of letters A = 5 inches

Hence , the equation is A = 5 inches

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The difference between the two populations will be 38466.22.

Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.

Given that Compare the January 2014 values of 5422.53 to the values of 43888.75 in 2010.

The population difference is calculated as:-

Differnce = 43888.75 - 5422.53

Difference = 38466.22

Therefore, the difference between the two populations will be 38466.22.

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The number of Gallons of paint that Daniel will need is; 3 gallons

The parameters given are;

Height of each wall = 78 inches = 78/12 = 6.5 ft

Length of wall 1 = 7 ft

Length of wall 2 = 6 ft

Length of wall 3 = 7 ft

Area of wall 1 = 6.5 * 7 = 45.5 sq.ft

Area of wall 2 = 6.5 * 6 = 39 sq.ft

Area of wall 3 = 6.5 * 7 = 45.5 sq.ft

Total area = 45.5 + 45.5 + 39

= 130 sq.ft

1 gallon of paint will cover 300 square feet of wall space

Number of gallons required = 300/130

= 2.3 gallons

Approximating to a whole number gives 3 gallons

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The total number of toys in Bashir's collection is 160.

A percentage is a number or ratio that can be expressed as a fraction of 100.

Let N represent the total number of toys in Bashir's collection

Since 136 of his toys are on the wall and 15% of the toys are in the display. We can write:

136 + (15% of N) = N

136 + (15/100 * N) = N

136 + 0.15N = N

N - 0.15N = 136

0.85N = 136

N = 136/0.85

N = 160

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The odds of drawing a 9 card or a black card from a standard 52- card deck is [tex]\frac{7}{13}[/tex]

A probability is a numerical representation of the likelihood or chance that a specific event will take place. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to describe probabilities.

A regular deck of cards contains 4 nines, therefore P(A) = [tex]\frac{4}{52}[/tex]

In a typical deck of cards, there are 26 black cards, hence P(B) = [tex]\frac{26}{52}[/tex].

When a card is both a 9 and black, there is overlap between the two events. Since there are only two such cards in a typical deck of cards, P(A and B) = [tex]\frac{2}{52}[/tex].

Using the P-formula (A or B),

P(A or B) = P(A) + P(B) (A and B)

P(A or B) = [tex]\frac{4}{52}+ \frac{26}{52} -\frac{2}{52}[/tex]

P(A or B) = [tex]\frac{28}{52} =\frac{7}{13}[/tex]

Therefore, the odds of drawing a 9 or a black card is [tex]\frac{7}{13}[/tex]

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For each of the functions, the constant c was determined so that the function would satisfy the conditions of being a pmf for a random variable X, and then a line graph was plotted for each pmf.

a. c = 4 so that f(x) = x/4;

Graph:

x f(x)

1 0.25

2 0.5

3 0.75

4 1

b. c = 1/30 so that f(x) = (1/30)x;

Graph:

x f(x)

1 0.03333

2 0.06667

3 0.1

4 0.13333

5 0.16667

6 0.2

7 0.23333

8 0.26667

9 0.3

10 0.33333

c. c = 1 so that f(x) = (1/4)x;

Graph:

x f(x)

1 0.25

2 0.125

3 0.0625

4 0.03125

d. c = 1/8 so that f(x) = (1/8)(x + 1)²;

Graph:

x f(x)

0 0.125

1 0.25

2 0.375

3 0.5

e. c = n so that f(x) = x/n;

Graph:

x f(x)

1 1/n

2 2/n

3 3/n

... ...

n 1

f. c = 1 so that f(x) = 1/(x + 1) − 1/(x + 2);

Graph:

x f(x)

0 1

1 0

2 −0.5

3 −0.333

4 −0.25

5 −0.2

... ...

For each of the functions, the constant c was determined so that the function would satisfy the conditions of being a pmf for a random variable X, and then a line graph was plotted for each pmf.

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Answer: Anand's salary after 10 years working for the company can be calculated as follows:

$60000 + $2000 * 10 = $60000 + $20000 = $80000.

To determine Anand's salary after t years working for the company, we can use the following formula:

$60000 + $2000 * t

So, if t is the number of years he has worked for the company, his salary after t years would be:

$60000 + $2000 * t.

Step-by-step explanation:

Answer: Let's call the height of the building "h".

The length of the shadow cast by the building is 5 feet, and the length of the shadow cast by the yardstick is also 5 feet. This means that the ratio of the height of the building to the length of its shadow is the same as the ratio of the height of the yardstick to the length of its shadow.

h / 5 = 1 / 5

Cross-multiplying, we get:

h = 5 * 1 / 5

h = 1

So the building is 1 foot tall.

Step-by-step explanation:

The area of the composite shapes found using the formula for the area of a rectangle and the perimeter of the figure indicates that the solution to the puzzle is; AGIH

The area of a rectangle = The length of the rectangle × The width of the rectangle

The area and perimeter of the shapes can be found by considering the shapes as composite figures as follows;

Area of shape 1 = Area of the 8 by 5 rectangle - Area of the 2 by 2 square at the top middle portion

Therefore;

The area of shape 2 = Area of the 13 by 4 rectangle - Area of the 2 by 3 rectangle at the top middle part of the shape.

(The width of the open part at the top of the rectangle is (13 cm - 2 × 5 cm) = 3 cm)

The four letter code is therefore;

AGIH

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Answer:The area of a circle can be calculated using the formula:

A = πr²

where A is the area and r is the radius.

Plugging in the given radius of 21 mm, we get:

A = π * 21² = π * 441

A = 441π mm²

To 1 decimal place, 441π can be approximated to 1393.1 mm². So, the area of the circle with a radius of 21 mm is approximately 1393.1 mm² to 1 decimal place.

Step-by-step explanation:

Answer:

Step-by-step explanation:

The expression (2x + 3y)² - 9 can be fully factorised as follows:

(2x + 3y)² - 9 = (2x + 3y + 3)(2x + 3y - 3)

= (2x + 3y + 3)(2x + 3y - 3)

The expression is now fully factorised.

a. Point P on the line and the direction vector v of the line can be expressed as P = (x0, y0, z0) + t(a, b, c).

b. The distance from the origin to the line -x=y+1, z=2 is the shortest distance between a point and a line.

Here (x0, y0, z0) is a point on the line and (a, b, c) is the direction vector. To find (x0, y0, z0) and (a, b, c), we need more information about the line.

This formula involves taking the dot product of the direction vector of the line and the difference between a point on the line and the origin. Again, we need more information about the line in order to calculate this distance.

Without more information about line L, it is not possible to find the point P or the distance from the origin to the line.

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The given  function  [tex]f(x)\left\{ \begin{array} { l } { \frac{1}{2}\left(x+4\right)-5 }\\{\ \ \ \ \ \ 3 } \end{array} \right.[/tex] can plotted on graph as given in the attachment.

Any relation in which each input has exactly one output is referred to as a function. To put it another way, the function outputs one value exactly for each input value. Because one is mapped to two different values, the graph above depicts a relation rather than a function. But if one were instead mapped to a single value, the relation above would become a function. Furthermore, output values could be identical to input values.

The function machine receives the input of the x-values. Following that, the function machine runs its operations and produces the y-values. Any function could be the one inside.

The given  function

So the graph of the above function will be plotted.

Hence The graph of the given function is potted below.

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Third side of a triangle is, 14.4

A triangle is a three sided polygon, which has three vertices and three angles which has the sum 180 degrees.

Given that;

Two sides of triangle are 8 and 12.

Now, By using Pythagoras theorem we get;

Let third side = x

⇒ x² = 8² + 12²

⇒ x² = 64 + 144

⇒ x² = 208

⇒ x = √208

⇒ x = 14.4

Thus, The third side = 14.4

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The probability of a defect is 5.7330 x [tex]10^{-5}[/tex] and the number of defects is 5.73.

To calculate the probability of a defect, we need to find the area under the standard normal curve that lies outside of the process control limits of 9.75 ounces and 10.25 ounces. We can use the standard normal distribution table to find this area.

First, we need to standardize the weight limits as follows -

[tex]Z_{lower}[/tex] = (9.75 - 10) / 0.25 = -4

[tex]Z_{upper}[/tex] = (10.25 - 10) / 0.25 = 4

Next, we will find the area under the standard normal curve that lies outside of these limits as follows -

P(Defect) = P(Z < -4) + P(Z > 4)

Using a standard normal distribution table, we can find that P(Z < -4) = 2.8665 x   [tex]10^{-5}[/tex]  and P(Z > 4) = 2.8665 x  [tex]10^{-5}[/tex] .

So, the total probability of a defect is -

P(Defect) = 2.8665 x  [tex]10^{-5}[/tex]  + 2.8665 x  [tex]10^{-5}[/tex]  = 5.7330 x  [tex]10^{-5}[/tex]

Finally, we will find the number of defects for a 1,000-unit production run as follows -

The number of defects = 1000 * 5.7330 x  [tex]10^{-5}[/tex]  = 5.73 (rounded to the nearest whole number).

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Answer:

x=2/3 or x= -3/10

Step-by-step explanation:

30x²-11x-6=0

ax²+bx+c=0

30x²+9x-20x-6=0

3x(10x+3)-2(10x+3)=0

(3x-2)(10x+3)=0

3x-2=0 or 10x+3=0

3x=2 or 10x= -3

x=2/3 or x= -3/10

The best buy is at the campus canteen. The solution has been obtained by using arithmetic operations.

What are arithmetic operations?

According to mathematicians, all real numbers may be described by the four basic operations, often known as "arithmetic operations." The phrases quotient, product, sum, and difference are produced by the four mathematical operations of division, multiplication, addition and subtraction, respectively.

We are given that a can of tab costs Rs. 7.20 at Timber City.

So, the cost of 6 packs will be

6 (7.20) = Rs. 43.20

But, on promotion, the cost is Rs. 39.60.

So, the discount is

D = Rs. 43.20 - Rs. 39.60

D = Rs. 3.60

Similarly,

A can of tab in the campus canteen costs Rs. 7.50.

So, the cost of 6 packs will be

6 (7.50) = Rs. 45

But, on promotion, the cost is Rs. 39.

So, the discount is

D = Rs. 45 - Rs. 39

D = Rs. 6

Hence, the best buy is at the campus canteen.

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The values of the expression following the collection of like terms, rearranging and factoring are presented as follows;

1.1 4·x + 5·a + 3·a + x - 6 = 5·x + 8·a - 6

1.2. 6·a·b + 2·b - 16·a - 2 = 2·(3·a·b + b - 8·a - 1)

1.3. 1024000 : 360000 : 180000 = 256 : 90 : 45

1.4. 150 min : 3 hours = 5 : 6

A mathematical expression comprises of mathematical symbols, numbers and variable, properly arranged to represent a value.

The simplification of the expressions can be presented as follows;

1.1. 4·x + 5·a + 3·a + (x - 6)

The terms in the above expression can be rearranged as follows;

4·x + 5·a + 3·a + (x - 6) = 4·x + x + 5·a + 3·a - 6

4·x + x + 5·a + 3·a - 6 = 5·x + 8·a - 6

Therefore, whereby the expression in the question is; 4·x + 5·a + 3·a + (x - 6), we get;

1.2. Whereby the possible expression in the question is; 6·a·b + 2·b - 16·a + (-2), we get;

The factor common to the terms of the expression 6·a·b + 2·b - 16·a + (-2) is 2. Therefore by factoring the expression, we get;

1.3. 1024000: 360,000 : 180,000

The expression is a ratio of the form a : b : c

Dividing each term in the ratio by 1,000, we get

1024000 ÷ 1000 : 360,000 ÷ 1000 : 180,000 ÷ 1000 = 1024 : 360 : 180

1024000: 360,000 : 180,000 = 1024 : 360 : 180

The common factors of 1024, 360 and 180 are; 1, 2, and 4

Therefore, the GCF of 1024, 360, and 180 is 4

Dividing the terms of the ratio 1024 : 360 : 180 by 4, we get;

1024 ÷ 4 : 360 ÷ 4 : 180 ÷ 4 = 256 : 90 : 45

Therefore;

1.4 150 min : 3 hours

The above ratio can be simplified by rewriting the ratio using unit conversion as follows;

3 hours = 180 minutes, therefore;

150 min : 3 hours = 150 min : 180 min

150 : 180 = 5 : 6

Therefore; 150 min : 180 min = 5 min : 6 min

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The range of the cigarette tax (in cents) for 50 selected states is 62.

The range of a function is defined as the set of all the possible output values that are valid for the given function.

The range of a given set of data is the difference between the largest value and the smallest value.

Given the data shown represent the cigarette tax (in cents) for 50 selected states

70,12,42,45,50,51,60,69,8,40,18

Required; The range

This is calculated as:

Range = highest - lowest

Range = 70 - 8

Range = 62

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